<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Examples</Text-field>

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This section contains some 21 examples illustrating the use of the Maple commands in Table 1. The organization of the examples is based on Table 3 and 4, and the remarks on nonlinear multivariate systems following Table 4.

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<Text-field style="Heading 2" layout="Heading 2">Linear Univariate Models</Text-field>

<Text-field style="Heading 3" layout="Heading 3">Overdetermined Case</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 1.

Fit 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 to ten data points along 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.JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Solution

Define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the fitting function.

Context Menu: Assign Function

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIj1GJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRj8vJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVi1JK211bmRlcm92ZXJHRiQ2Jy1GQzYtUSYmU3VtO0YnRkAvRkdRJnVuc2V0RicvRklGam4vRktGMS9GTUZqbi9GT0YxL0ZRRjEvRlNGam4vRlVRJjAuMGVtRicvRlhRLDAuMTY2NjY2N2VtRictRiM2KC1GLDYlUSJrRidGL0YyRkItSSNtbkdGJDYkUSIxRidGQEY9LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRkAtRiM2Ji1GW3A2JFEiNUYnRkBGPUZecEZARlIvJSxhY2NlbnR1bmRlckdGPy1JJW1zdWJHRiQ2JS1GLDYlUSJiRidGL0YyLUYjNiZGZ29GPUZecEZALyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRkM2LVEifkYnRkBGRkZIRkpGTEZORlBGUkZhby9GWEZiby1GLDYlUSRzaW5GJy9GMEY/RkAtRjY2JC1GIzYoRmdvRmNxRjpGPUZecEZARkAtRiw2I1EhRidGPUZA

Ly1JImZHNiI2I0kieEdGJSwsKiYmSSJiR0YlNiMiIiJGLS1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRi1GLSomJkYrNiMiIiNGLS1GLzYjLCQqJkY2Ri1GJ0YtRi1GLUYtKiYmRis2IyIiJEYtLUYvNiMsJComRj5GLUYnRi1GLUYtRi0qJiZGKzYjIiIlRi0tRi82IywkKiZGRkYtRidGLUYtRi1GLSomJkYrNiMiIiZGLS1GLzYjLCQqJkZORi1GJ0YtRi1GLUYt

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEzYXNzaWdufmFzfmZ1bmN0aW9uRidGL0YyRkM=

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

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Define a list of x-values (as floats).

Form a list of corresponding y-values: 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Apply the LeastSquares command from the CurveFitting package.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCwqJl00MDBFMTdERDIwNEE2Njk2IiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRitGJUYlKiZdNDAwOTg4QUE1M0E4QjVCOUYlLUYnNiMsJComJCIiIyIiIUYlRi1GJUYlRiUhIiIqJl00MDAwOUNGMkEyNjNDNTMxRiUtRic2IywkKiYkIiIkRjZGJUYtRiVGJUYlRiUqJl0zRkZBQ0IwMkVEMDU0MjJGRiUtRic2IywkKiYkIiIlRjZGJUYtRiVGJUYlRjcqJl0zRkY1RUI0RkM1ODRDM0U3RiUtRic2IywkKiYkIiImRjZGJUYtRiVGJUYlRiU=

Apply the LeastSquares command from the LinearAlgebra package.

The arguments are a set of equations of the form 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, and a set of parameters (the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn).

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCcvJkkiYkc2IjYjIiIiXTQwMEUxN0REMjA0QTY2OUEvJkYlNiMiIiNdQzAwOTg4QUE1M0E4QjVCQS8mRiU2IyIiJF00MDAwOUNGMkEyNjNDNTMyLyZGJTYjIiIlXUJGRkFDQjAyRUQwNTQyMzAvJkYlNiMiIiZdM0ZGNUVCNEZDNTg0QzNFRg==

Notice that the output is not the fitting function, but a set of equations defining the parameters.

To the problem in the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, apply the LeastSquares command from LinearAlgebra.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

The output is now a vector of values for the parameters.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Apply the LSSolve command from the Optimization package. The input is the list 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.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

The output is a list, the first member of which is half the sum of the squares of the deviations; and the second of which is a vector of values for the parameters.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Apply the LinearFit command from the Statistics package.

The arguments are a list of basis functions for the linear model, the data, and the independent variable for the model.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

In LinearFit (algebraic form)

SVD tolerance set to .10e-11 confidence level set to .95 final value of residual sum of squares: 30.8945581508559

LCwqJl00MDBFMTdERDIwNEE2Njk4IiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRitGJUYlKiZdNDAwOTg4QUE1M0E4QjVCOUYlLUYnNiMsJComIiIjRiVGLUYlRiVGJSEiIiomXTQwMDA5Q0YyQTI2M0M1MzFGJS1GJzYjLCQqJiIiJEYlRi1GJUYlRiVGJSomXTNGRkFDQjAyRUQwNTQyMzBGJS1GJzYjLCQqJiIiJUYlRi1GJUYlRiVGNSomXTNGRjVFQjRGQzU4NEMzRUZGJS1GJzYjLCQqJiIiJkYlRi1GJUYlRiVGJQ==

By setting infolevel to 5, additional information about the calculation is printed. The fitting function is returned.

Apply the Fit command from Statistics; the first argument must now be the model function.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

In Fit

In LinearFit (algebraic form) SVD tolerance set to .10e-11 confidence level set to .95 final value of residual sum of squares: 30.8945581508559

Notice how Fit passed the problem off to LinearFit.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Apply the Fit command from Statistics so as to return m, a module whose exports are the entries of Table 2.

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

Access the exports of module LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= singly.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzotRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEoUmVzdWx0c0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRNXJlc2lkdWFsc3Vtb2ZzcXVhcmVzRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjlGWkY5

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

XTQwM0VFNTAxQzM1MjRCRDU=

Not recommended: Return all 16 exports in a (poorly formatted) list: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzotRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEoUmVzdWx0c0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiNRIUYnRjlGOUY5

The following device line-breaks the exports, making them easier to read.

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbXNHRiQ2I1EzcmVzaWR1YWxtZWFuc3F1YXJlRictSSNtb0dGJDYtUSI9RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtbkdGJDYkUTE2LjE3ODkxMTYzMDE3MTE4RidGMy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJSlyZWFkb25seUdGOEYz

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbXNHRiQ2I1E6cmVzaWR1YWxzdGFuZGFyZGRldmlhdGlvbkYnLUkjbW9HRiQ2LVEiPUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbW5HRiQ2JFExMi40ODU3NDE2NjYwMTY2NEYnRjMvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRjhGMw==

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

NyI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Underdetermined Case</Text-field>

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 2.JSFH

Fit 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 to the data points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiNUYnRjRGNEY0RjQ= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ=.

JSFH

Solution

Initialize Maple.

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

Define the fitting function: Context Menu: Assign Function

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

Ly1JImZHNiI2I0kieEdGJSwoKiZJImFHRiUiIiIpRiciIiNGK0YrKiZJImJHRiVGK0YnRitGK0kiY0dGJUYr

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJmRzYi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

This is an interpolation problem requiring the solution of two consistent equations in three unknowns. It is not necessarily a least-squares problem.

Write and solve two (consistent) equations in three unknowns.

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print(); # input placeholder

PCUvSSJhRzYiLCYjIiIoIiIjISIiKiYjIiIiRilGLUkiY0dGJUYtRi0vSSJiR0YlLCYjIiM8RilGLSomIyIiJEYpRi1GLkYtRiovRi5GLg==

Obtain the general solution, a one-parameter family of interpolating functions.

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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JSFH

Use the LeastSquares command from the CurveFitting package:

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print(); # input placeholder

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Although the calculation is passed off to the LeastSquares command in LinearAlgebra, which has provision for controlling the name of the free parameter, this control is lacking in the CurveFitting package.

Use the LeastSquares command from LinearAlgebra. The arguments here will be a set of equations and a set of parameters. Note the control over the free parameter.

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print(); # input placeholder

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The LeastSquares command in LinearAlgebra returns a set of equations defining the parameters, which then have to be transferred to the model to obtain the fitting function. The appearance of the free parameter is best explained via the matrix formulation of the problem.

Cast the problem in the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==.

Convert equations to matrix/vector form.

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print(); # input placeholder

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Obtain the general solution.

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print(); # input placeholder

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

Obtain the minimum-norm solution.

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print(); # input placeholder

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Project V onto the row space of A.

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print(); # input placeholder

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

The numeric solvers of the Optimization package, being local, will not necessarily find the minimum-norm solution, and might return any member of the general solution. The numeric solvers of the Statistics package reject underdetermined problems.JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

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Example 3.

Fit 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 to the data point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiNUYnRjRGNEY0RjQ= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ=.

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Solution

Initialize Maple.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0YzUSdub3JtYWxGJw==

Define the fitting function: Context Menu: Assign Function

Ly1JImZHNiI2I0kieEdGJSwoKiZJImFHRiUiIiIpRiciIiNGK0YrKiZJImJHRiVGK0YnRitGK0kiY0dGJUYr

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJmRzYi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Use the LeastSquares command from the CurveFitting package:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCwiIiUiIiImSSNfdEc2IjYjIiIjISIiJkYmNiMiIiRGKiomRiVGJEkieEdGJ0YkRiQqJkYrRiQpRi9GKUYkRiQ=

Use the LeastSquares command from LinearAlgebra. The arguments here will be a set of equations and a set of parameters.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCUvSSJhRzYiLCgiIiUiIiImSSJzR0YlNiMiIiMhIiImRio2IyIiJEYtL0kiYkdGJUYpL0kiY0dGJUYu

LCgqJiwoIiIlIiIiJkkic0c2IjYjIiIjISIiJkYoNiMiIiRGLEYmKUkieEdGKUYrRiZGJiomRidGJkYxRiZGJkYtRiY=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

Obtain the general solution

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEqUGFyYW1WZWN0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEuTGluZWFyQWxnZWJyYUYnRi9GMi1GNjYtUSM6LUYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRLUxlYXN0U3F1YXJlc0YnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYrLUYsNiVRIkFGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZVL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNidRInZGJy8lJWJvbGRHRjEvRjBGPS9GM1ElYm9sZEYnLyUrZm9udHdlaWdodEdGam9GXW8tRiw2JVElZnJlZUYnRi9GMi1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GLDYlUSJzRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiO0YnRjlGO0Zgb0ZARkJGREZGRkhGVUZNLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlYvJSZkZXB0aEdGYHEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZVRlctRiw2JVElZXZhbEYnRi9GMi1GZm42JC1GIzYpLUYsNiVRImZGJ0YvRjItRmZuNiQtRiM2JS1GLDYlUSJ4RidGL0YyRmZwRjlGOUZdby1GLDYlUSdFcXVhdGVGJ0YvRjItRmZuNiQtRiM2Jy1GZm42Ji1GIzYpLUYsNiVRImFGJ0YvRjJGXW8tRiw2JVEiYkYnRi9GMkZdby1GLDYlUSJjRidGL0YyRmZwRjlGOS8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGXW9GK0ZmcEY5RjlGZnBGOUY5RmZwRjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

LCgqJiwoIiIlIiIiJkkic0c2IjYjIiIjISIiJkYoNiMiIiRGLEYmKUkieEdGKUYrRiZGJiomRidGJkYxRiZGJkYtRiY=

Obtain the minimum-norm solution

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

The null space of A is spanned by the vectors 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.

JSFHJSFH

<Text-field style="Heading 2" layout="Heading 2">Nonlinear Univariate Models</Text-field>

<Text-field style="Heading 3" layout="Heading 3">Overdetermined Case</Text-field>

<Text-field style="Heading 4" layout="Heading 4">Logarithmic Model</Text-field>

JSFH

Example 4.

Fit 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 to the 45 data points shown in Figure 1.

X:=[.4415800447, .5203850932, .5586711668, .6765059811, .7450887166, .9557085792, .9581748469, 1.056473538, 1.566338190, 1.816703465, 2.388745916, 2.650935552, 2.961694105, 3.135006301, 3.413191570, 3.525872385, 4.033977197, 4.058984896, 4.087467046, 4.448111434, 4.478647296, 4.704561792, 4.927323253, 4.937647961, 5.212111955, 5.350970947, 6.569312236, 6.619738280, 6.636234495, 7.225453222, 7.321555106, 7.421499760, 7.628867977, 7.696458584, 7.869296084, 8.005454234, 8.192046666, 8.247384597, 8.319562068, 8.352030955, 9.377657024, 9.532776673, 9.722178164, 9.759535988, 9.887836167]:

Y:=[-.3165315447, 0.525739071e-1, .2280755754, .9930696348, 1.005519640, 2.050502588, 1.684642486, 3.030733346, 2.676977277, 5.307499018, 4.151074969, 6.894632989, 4.731555880, 7.599050973, 3.409766036, 6.936460408, 4.947406689, 6.823078626, 4.979021138, 7.125182710, 4.548574222, 9.303837054, 4.070632602, 7.469734020, 4.867068792, 8.438200908, 4.588336461, 9.204201002, 7.677634124, 10.31267885, 5.580727779, 8.814457831, 4.857491046, 10.55896517, 4.913343507, 8.240369267, 6.647593036, 9.162657236, 8.355828851, 9.204265632, 7.843490860, 9.640628922, 5.293937431, 12.36862840, 5.324349598]:

plot(X,Y,style=point);

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

Figure 1 Data points to be fitted with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJhRidGL0YyLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVi1GLDYlUSJiRidGL0YyLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GaW4tRiw2JVEjbG5GJy9GMEY9RjktSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyRjlGOUY5. (Data hidden behind table: lists X and Y of values of the independent and dependent variables, respectively.)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Solution

Define f, the logarithmic model function.

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

Form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, the sum of squares of deviations:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

Apply the LogarithmicFit command from Statistics

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQsJl00MDAxNzc1NEJCMkNBNTk0IiIiKiZdNDAwNURGNzhGMzU4N0NEQ0YlLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGLUYlRiVdNDA2MTRCMjMwNzA2QjM3Rg==

Apply the NonlinearFit command from Statistics

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQsJl00MDAxNzc1NEJDNUNDMkM0IiIiKiZdNDAwNURGNzhGMzRDOTNDNkYlLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGLUYlRiUkIitlLVskUSIhIig=

Minimize LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn via the Optimization package

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIi9GNmUtWyRRIiEjNjckL0kiYUc2Il00MDAxNzc1NEJCMkNBNTlFL0kiYkdGKV00MDA1REY3OEYzNTg3Q0Q3

Form and solve the normal equations, then evaluate the resulting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEnUGFyYW1zRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiw2JVEnZnNvbHZlRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiYtRlM2Ji1GIzYoLUYsNiVRJWRpZmZGJ0YvRjItRlM2JC1GIzYmLUYsNiVRI1NTRidGL0YyLUY2Ni1RIixGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktRJjAuMGVtRicvRk5RLDAuMzMzMzMzM2VtRictRiw2JVEiYUYnRi9GMkY5RjlGX29GZW4tRlM2JC1GIzYmRlxvRl9vLUYsNiVRImJGJ0YvRjJGOUY5RjlGOS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGX28tRlM2Ji1GIzYmRmdvRl9vRl5wRjlGOUZhcEZkcEY5RjktRjY2LVEiO0YnRjlGO0Zib0ZARkJGREZGRkhGY29GTS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zkby8lJmRlcHRoR0ZjcS8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GX3E2JkZhcUZkcUZmcS9GaXFRJWF1dG9GJy1GLDYlUSVldmFsRidGL0YyLUZTNiQtRiM2JkZcb0Zfb0YrRjlGOS8lK2V4ZWN1dGFibGVHRj1GOQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCQvSSJhRzYiJCIrLHJFJD0jISIqL0kiYkdGJSQiKyx1Nk1GRig=

JCIrZC1bJFEiISIo

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The solution obtained by the linearization in LogarithmicFit closely matches the solutions obtained by methods that do not linearize.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Power Model</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH

Example 5.

Fit 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 to the 45 data points shown in Figure 2.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

X:=[.3311845858, .3382461426, .3596315385, .4171713246, .4172203361, .4373441686, .4871624354, .5074279163, .5970838801, .6782429163, .6822631566, .6857075734, .7123173532, .7928747878, .8613884024, .9423775810, 1.013359430, 1.015837175, 1.039779012, 1.073557773, 1.087726149, 1.104538148, 1.183382262, 1.288303209, 1.371372688, 1.411264277, 1.510435824, 1.541170242, 1.721992776, 1.884733992, 1.890335165, 1.935953579, 1.943825167, 2.302765894, 2.439615410, 2.443593963, 2.554861738, 2.652786753, 2.773245229, 2.775144735, 2.868025527, 2.888689619, 2.940485400, 2.987329661, 2.996133391]:

Y:=[3.901426454, 5.552778517, 3.682748534, 4.425793308, 2.581500466, 3.925060388, 3.308707232, 4.501901432, 2.869493552, 2.887041285, 1.829623341, 3.125460247, 1.775237930, 2.823371431, 2.220192758, 2.501806865, 1.981506660, 2.571558903, 1.946127380, 2.664282284, 1.885672629, 2.425196004, 1.599871000, 1.675009728, 1.282658772, 1.885755564, .8991034242, 1.773028886, 1.093699604, 1.411716974, .8965011818, 1.511418676, 1.255943074, 1.115462934, 1.071285035, 1.070063783, .7260572291, 1.111299304, .8814155598, 1.272545739, .5739487804, 1.237321990, .8460171871, 1.022644001, .9277633116]:

plot(X,Y,style=point);

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

Figure 2 Data points to be fitted with 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. (Data hidden behind table: lists X and Y of values of the independent and dependent variables, respectively.)

JSFH

Solution

Context Menu: Assign Function

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

Ly1JImdHNiI2I0kieEdGJSomSSJhR0YlIiIiKUYnSSJiR0YlRio=

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJnRzYi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

Apply the PowerFit command from Statistics

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

Evaluate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn for the linearized fit

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

XTQwMjNEQTdFM0NENkI0RTA=

Apply the NonlinearFit command from Statistics

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQsJComXTQwMDA4MkNCRkVBREVFNTUiIiIpSSJ4RzYiXTNGRTZDODg5OUZGMTc3MUIhIiJGJiQiK2sjKilcIykqISIq

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjNVIlInUmsjKilcIykqISM8NyQvSSJhRzYiXTQwMDA4MkNDMEM2NkY4M0IvSSJiR0YpXUJGRTZDODg5N0JBNEJFOEY=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCQvSSJhRzYiJCIrX2MnUTEjISIqL0kiYkdGJSQhKyIqb3o+ciEjNQ==

JCIrZyMqKVwjKSohIio=

JSFH

The parameters computed by the linearization in PowerFit differ slightly from those computed by the other methods which don't linearize. The sum of squares of residuals returned by PowerFit is for the linearized model, not the nonlinear model; when corrected for the linearization, it is slightly larger than the value for the nonlinear fits.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Exponential Model</Text-field>

JSFH

Example 6.

Fit 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 to the 45 data points shown in Figure 3.

JSFH

X:=[.3249060766, .4638590607, .4644519699, .4646954321, .5512620502, .5958265967, .6201785971, .6231171221, .6302313432, .6306638781, .6993610748, .7318673274, .8091876437, .9008821207, .9849002264, 1.064226572, 1.110383885, 1.259818068, 1.435261400, 1.453715259, 1.482196702, 1.532050477, 1.581696829, 1.712052277, 1.870325124, 1.872936109, 2.060662252, 2.074651185, 2.085638251, 2.114074877, 2.319376166, 2.357983108, 2.409166844, 2.417540344, 2.503005511, 2.523280179, 2.598296000, 2.603682807, 2.644623630, 2.662815612, 2.797604341, 2.896353971, 2.921562129, 2.927435786, 2.966131074]:

Y:=[1.593149583, 1.590034983, 1.011420580, 2.022496449, 1.223729582, 1.449732045, .9069637640, 1.422300135, 1.157919343, 1.543425077, 1.103220812, 1.437868212, .7945660884, 1.064526071, .9033515451, .9495079084, .8273875646, 1.159219310, .7323213934, 1.012091251, .5669217104, .8212180644, .5287792925, .6033309380, .3240342593, .5390709438, .3781516600, .6085085079, .3715978170, .5464110888, .2366337341, .3838739654, .3703637640, .4050191695, .3121357178, .3419301014, .1946629567, .3555389028, .1884514276, .4031447974, .2821896664, .3160107514, .2328625706, .2576745642, .1755520822]:

plot(X,Y,style=point);

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

Figure 3 Data points to be fitted with 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. (Data hidden behind table: lists X and Y of values of the independent and dependent variables, respectively.)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Solution

Context Menu: Assign Function

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

Ly1JImhHNiI2I0kieEdGJSomSSJhR0YlIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkkiYkdGJUYqRidGKkYq

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJoRzYi

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Apply the ExponentialFit command from Statistics

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print(); # input placeholder

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElZXZhbEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRI1NTRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNjYmLUYjNi0tRiw2JVEiYUYnRi9GMi1GPjYtUSI9RidGQUZDL0ZHRkVGSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRl5vLUYjNiUtRiw2I1EhRictSSVtc3VwR0YkNiUtRj42LVEvJkV4cG9uZW50aWFsRTtGJ0ZBL0ZEUSZ1bnNldEYnL0ZHRlxwL0ZJRlxwL0ZLRlxwL0ZNRlxwL0ZPRlxwL0ZRRlxwRlIvRlZGVC1GIzYnRmJvLUYjNiYtRiw2JVEiU0YnRi9GMi1GNjYmLUYjNiQtSSNtbkdGJDYkUSIzRidGQUZBRkEvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2Ji1GIzYkLUZgcTYkUSIxRidGQUZBRkFGY3FGZnFGQUZib0YvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkFGPS1GLDYlUSJiRidGL0YyRmluRmhwLUY2NiYtRiM2JUZfcS8lK2V4ZWN1dGFibGVHRkVGQUZBRmNxRmZxLUY2NiYtRiM2JS1GYHE2JFEiMkYnRkFGanJGQUZBRmNxRmZxRmpyRkFGQUZjcUZmcUZqckZBRkFGanJGQQ==

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print(); # input placeholder

XTNGRjc3MDUzQjU5RjE3ODk=

Apply the NonlinearFit command from Statistics

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print(); # input placeholder

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print(); # input placeholder

NyQkIjNkIlIqM0BGLjVOISM7NyQvSSJhRzYiXUMwNEE0OEI0OTI5NDMxNjIvSSJiR0YpXUMwNjE1QTU0MDU3NEMxRTY=

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print(); # input placeholder

PCQvSSJhRzYiJCIreUsoKlI/ISIqL0kiYkdGJSQhK09OKHlCKCEjNQ==

JCIrQk4mNFgiISIq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The parameters computed by the linearization in ExponentialFit differ slightly from those computed by the other methods which don't linearize. The sum of squares of residuals returned by ExponentialFit is for the linearized model, not the nonlinear model; when corrected for the linearization, it is slightly larger than the value for the nonlinear fits.

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<Text-field style="Heading 4" layout="Heading 4">Michaelis-Menten Model</Text-field>

Example 7.

Fit 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, the Michaelis-Menten model, to the 46 data points shown in Figure 4.

JSFH

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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

Figure 4 Data points to be fitted with 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. (Data hidden behind table: lists S and V of values of the independent and dependent variables, respectively.)

This example is a summary of one that appears in the ebook, Advanced Engineering Mathematics with Maple, an example that also appears in the Reporter article Nonlinear Fit, Optimization, and the DirectSearch Package. The fit is obtained with two different linearizations, and nonlinearly, the outcome being that the linearized solutions provide decidedly poor fits.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Solution

Define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInNGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==: Context Menu: Assign Function

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

Ly1JInZHNiI2I0kic0dGJSooSSJhR0YlIiIiRidGKiwmSSJiR0YlRipGJ0YqISIi

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJ2RzYi

Define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, sum of squares of deviations.

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

Linearization 1

Linearization 2

The linearization 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 requires the computation of the reciprocals 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 and 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.

The linearization 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 requires a list of ratios 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 with new independent variable 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.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Obtain the least-squares regression line 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

Here, 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 and 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.

Here, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRIkFGJ0Y0RjcvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJ0Y+RitGV0Y+ and 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.

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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print(); # input placeholder

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Nonlinear Fit

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQsJCooXTQwMEJEMjU4QzRENzA4MDYiIiJJInNHNiJGJiwmXTQwMjA3MzAyNEUwM0QxMzdGJkYnRiYhIiJGJiQiKzYneURpIiEiKg==

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LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjArXnAwJCpHNikhIzo3JC9JImFHNiJdNDAwQkQyNThDNkREQjlCNS9JImJHRildNDAyMDczMDI1MEREQkQyQg==

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCQvSSJhRzYiJCIrciMzeFokISIqL0kiYkdGJSQiKyxxaUMjKUYo

Sum of Squares for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

XTQwMTlCQTQ4OTE4QjIxOTU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Sum of Squares for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+:

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print(); # input placeholder

XTQwMTIxQjhCNzE5QjUzNzM=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Sum of Squares for nonlinear fitting function:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEkU29sRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGPkY+LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkBGPg==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrNid5RGkiISIq

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The NonlinearFit command from Statistics can return the fitting function, but the Statistics package's LSSolve command, whose input is a list of deviations (called residuals) returns the parameter values. In this example, it is even possible to form the normal equation and solve them numerically.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Figure 5 compares the graphs of the three fitting functions. Both from the graph and from the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, it should be clear that linearizations do not necessarily provide the best fits to data.

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Figure 5 Nonlinear fit (black), LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ (red), LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ (green) superimposed on Figure 4

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Underdetermined Case</Text-field>

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 8.

Fit 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 to the two points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0RjQ=.

This problem is essentially an interpolation, with the expected result being a one-parameter family of curves all going through the two given points. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= were a linear function, such an outcome, and the means to achieve it, would be clear. For this particular LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, it is possible to find this one-parameter family of solutions, but in general, it might not be possible to implement the requisite manipulations.

Exact Solution

Restart Maple.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

Control-drag the equation 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 Context Menu: Assign Function

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ki1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieEYnRjRGNy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnL0Y4USdub3JtYWxGJ0ZILUkjbW9HRiQ2LVEiPUYnRkgvJSZmZW5jZUdGRC8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zobi1GIzYoLUklbXN1cEdGJDYlLUZLNi1RLyZFeHBvbmVudGlhbEU7RidGSEZORlBGUkZURlZGWEZaL0ZnblEmMC4wZW1GJy9Gam5RLDAuMTExMTExMWVtRictRiM2KC1GLDYlUSJhRidGNEY3LUZLNi1RMSZJbnZpc2libGVUaW1lcztGJ0ZIRk5GUEZSRlRGVkZYRlpGY28vRmpuRmRvRj9GQkZFRkgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRlxwLUY7NiQtRiM2KS1GLDYlUSJiRidGNEY3LUZLNi1RIitGJ0ZIRk5GUEZSRlRGVkZYRlovRmduUSwwLjIyMjIyMjJlbUYnL0ZqbkZecS1GIzYoLUYsNiVRI2xuRicvRjVGREZILUZLNi1RMCZBcHBseUZ1bmN0aW9uO0YnRkhGTkZQRlJGVEZWRlhGWkZjb0ZfcC1GOzYkLUYjNihGKy1GIzYoLUYsNiVRImNGJ0Y0RjdGanBGP0ZCRkVGSEYrRkJGRUZIRkhGQkZFRkhGK0ZCRkVGSEZIRkJGRUZIRitGQkZFRkhGK0ZCRkg=

Ly1JImZHNiI2I0kieEdGJSomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJkkiYUdGJSIiIkYnRjFGMSwmSSJiR0YlRjEtSSNsbkdGKzYjLCZJImNHRiVGMUYnRjFGMUYx

proc(g) assign( op( 0, lhs(g) ) = unapply( rhs(g), op( lhs(g) ) ) ); op( 0, lhs(g) ) end proc( '' );

SSJmRzYi

Solve for b and c as functions of a.

Obtain the one-parameter family of solutions.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJImFHRikiIiJJInhHRilGLUYtLCotSSNsbkdGJjYjIiIjRi0tRjE2IyokLCZGLSEiIi1GJTYjLCQqJiIiJEYtLUYlNiMsJEYsRjhGLUY4Ri1GOEYtRjxGOC1GMTYjKiYsKkY9Ri1GOUY4Ri5GOComRi5GLUY5Ri1GLUYtRjdGOEY4Ri1GOA==

JSFH

The LSSolve command in the Optimization package requires at least as many residuals as parameters; otherwise, an error results. Hence, it really does not apply here.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 9.

Fit 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 to the two points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiNUYnRjRGNEY0RjQ=.

The equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIjNGJ0Y+Rj4= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIjVGJ0Y+Rj4= would necessarily be inconsistent, so this is not an interpolation, but a problem of fitting by least squares. A general solution consisting of a two-parameter family of curves is expected.

General Solution

If necessary, define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

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

Define S the sum of squares.

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

Obtain and solve the normal equations. The parameters a and b are free, and 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.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCUvSSJhRzYiRiQvSSJiR0YlRicvSSJjR0YlLCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiYsJiomLUYsNiNGJCIiIkYnRjdGNyIiJSEiIkY3RjVGOUY5RjdGN0Y5

Evaluate the sum of squares for this solution.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Obtain the general solution to the underdetermined, inconsistent least-squares problem.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

KiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSJhR0YoIiIiSSJ4R0YoRixGLCwmSSJiR0YoRiwtSSNsbkdGJTYjLCgtRiQ2IywkKiYsJiomLUYkNiNGK0YsRi9GLEYsIiIlISIiRixGOkY9Rj1GLEYsRj1GLUYsRixGLA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Casual inspection shows that every member of the general solution passes through LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiNEYnRjRGNEY0RjQ=.

As in Example 8, the LSSolve command in the Optimization package requires at least as many residuals as parameters; otherwise, an error results. Hence, it really does not apply here.

<Text-field style="Heading 2" layout="Heading 2">Linear Multivariate Models</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

This section considers linear multivariate models that are cast in the form LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==. As per Table 4, the examples are classified by the properties of the r \303\227 c matrix A, and the vector v.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Initialize Maple.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3"><Equation executable="false" style="Heading 3" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPEYnRi8vJSdpdGFsaWNHRjEvRjNRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTS1GLDYlUSJjRidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YvRjlGOw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPEYnRi8vJSdpdGFsaWNHRjEvRjNRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTS1GLDYlUSJjRidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YvRjlGOw==</Equation></Text-field>

<Text-field style="Heading 4" layout="Heading 4">Full Rank</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Define the full-rank matrix

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

for which

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI0FmRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIk

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

Example 10.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNBZkYnRi9GMkY5, and 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Since there are fewer equations than variables, the system LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA== cannot be inconsistent. Each row in A represents the left-hand side of a distinct equation, so no matter what appears on the right, the equations in the system must be consistent.

Solution

Apply the LeastSquares command, designating s as the free variable base-name. The result is the general solution containing one free parameter. The solution is u, a vector of parameter values.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1ElVWdlbkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSomY29sb25lcTtGJ0YvRjVGOC8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tRiw2JVEtTGVhc3RTcXVhcmVzRicvRjNGMS9GNlEnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ky1GLDYlUSNBZkYnRlRGVS1GOzYtUSIsRicvRjZRJ25vcm1hbEYnRj4vRkFGMUZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRictRlg2Ji1GIzYpLUkjbW5HRiQ2JFEiMUYnRlxvRmluLUZobzYkUSIyRidGXG9GaW4tRmhvNiRRIjNGJ0Zcby8lK2V4ZWN1dGFibGVHRjRGXG9GXG8vJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRmluLUYsNiVRJWZyZWVGJ0ZURlUtRjs2LVEiPUYnRlxvRj5GQEZCRkRGRkZIRkpGTEZPLUYsNiVRInNGJ0ZURlVGYXBGXG9GXG9GYXBGXG8=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1eVs2OC0zV25XPQ==

Obtain the minimum-norm least-squares solution. This is the projection of the general solution onto the row space of Af.

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1VSJwV0AhM1duVz0=

Obtain P, the matrix that projects onto the row space of A.

The columns of N are a basis for the row space.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Project LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EiVUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctRiw2J1EkZ2VuRidGL0YyRjVGOC9GNlEnbm9ybWFsRic= onto the row space. The result is the minimum-norm solution, a vector that lies in the row space of Af.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYnUSVVZ2VuRicvJSVib2xkR0YxL0YwRj0vRjNRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlYvJStleGVjdXRhYmxlR0Y9Rjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1eSNwV0AhM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Rank-Deficient</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Define the rank-deficient matrix

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

for which

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI0FkRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIj

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH

Example 11.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNBZEYnRi9GMkY5, and 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.

JSFH

That the system is consistent can be seen from

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRjY2Ji1GIzYnLUYsNiVRI0FkRidGL0YyLUkjbW9HRiQ2LVEifGdyRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlgtRjY2Ji1GIzYpLUkjbW5HRiQ2JFEiOUYnRkUtRkI2LVEiLEYnRkVGRy9GS0YxL0ZNRklGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWlEsMC4zMzMzMzMzZW1GJy1Gam42JFEiM0YnRkVGXW8tRmpuNiRRIjRGJ0ZFLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRlxwRkVGRUZecEZhcEZccEZFRkVGXHBGRQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNTFJZTctM1duVz0=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

General solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1TThaOS0zV25XPQ==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Minimum-norm solution:

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1cTlaOS0zV25XPQ==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

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Example 12.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNBZEYnRi9GMkY5, and 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

That the system is inconsistent can be seen from

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNWVKZTctM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

General solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1VTxaOS0zV25XPQ==

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Minimum-norm solution:

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1eT1aOS0zV25XPQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

<Text-field style="Heading 3" layout="Heading 3"><Equation executable="false" style="Heading 3" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPUYnRi8vJSdpdGFsaWNHRjEvRjNRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTS1GLDYlUSJjRidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YvRjlGOw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPUYnRi8vJSdpdGFsaWNHRjEvRjNRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTS1GLDYlUSJjRidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YvRjlGOw==</Equation></Text-field>

<Text-field style="Heading 4" layout="Heading 4">Full Rank</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

A full-rank matrix in a square system must necessarily be consistent, and therefore have a unique solution. There cannot be a least-squares problem in this case.

JSFHJSFH

<Text-field style="Heading 4" layout="Heading 4">Rank-Deficient</Text-field>

JSFH

Define the rank-deficient matrix

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

JSFH

for which

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI0JkRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIj

JSFHJSFH

JSFH

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

Example 13.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNCZEYnRi9GMkY5, and 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.

JSFH

That the system is consistent can be seen from

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNTVMZTctM1duVz0=

JSFH

General solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1OT9aOS0zV25XPQ==

JSFH

Minimum-norm solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1XUBaOS0zV25XPQ==

evalf[10]( );

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1J0dzV0AhM1duVz0=

JSFH

Because this system is essentially just an underdetermined one, a general solution is available with the LinearSolve command in the LinearAlgebra package.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1JXBzV0AhM1duVz0=

JSFH

Seek a least-squares solution numerically:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1bUhaOS0zV25XPQ==

That this is a member of the general solution can be seen by projecting it onto the row space of Bd.

The columns of N are a basis for the row space.

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

Project LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EjVW5GJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3L0Y2USdub3JtYWxGJw== onto the row space. The result is the minimum-norm solution, a vector that lies in the row space of Bd.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYnUSNVbkYnLyUlYm9sZEdGMS9GMEY9L0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZWLyUrZXhlY3V0YWJsZUdGPUY5

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1XVpaOS0zV25XPQ==

JSFH

However, to obtain the minimum-norm solution numerically, specify that the calculation is to be based on the singular value decomposition, rather than on the default QR decomposition.JSFH

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1JylbWjktM1duVz0=

JSFH

<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 14.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNCZEYnRi9GMkY5, and 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.

JSFH

That the system is inconsistent can be seen from

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNWlNZTctM1duVz0=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

General solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1QV1aOS0zV25XPQ==

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Minimum-norm solution:

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1ZV5aOS0zV25XPQ==

evalf[10]( );

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1JUh2V0AhM1duVz0=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Seek a least-squares solution numerically:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1bWJaOS0zV25XPQ==

JSFH

The default method, based on a QR decomposition, utterly fails because this decomposition does not have an efficient way to determine rank. For problems such as this (and Example 13), specify the method as the one based on the singular value decomposition.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1LWRaOS0zV25XPQ==

JSFH

Alternatively, use the LinearFit command from the Statistics package. Although this command is based on the LeastSquares from LinearAlgebra, there is an additional wrapper that attempts to deal with the issues raised by numeric calculations.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Warning, model is not of full rank

LCoqJl0zRjgyMTE3RkRBOUIwQTY4IiIiSSJ4RzYiRiVGJSomXTNGNzBGQUQ2NDhGNUU1NkNGJUkieUdGJ0YlISIiKiZdM0Y2RUZGQzU4QTYzMTIyNkYlSSJ6R0YnRiVGKyomXTNGOEVGQTM4NzkzNEYxRTRGJUkid0dGJ0YlRiU=

JSFH

The deficiency in rank of the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQmRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn has been detected, and the calculation is based on the singular value decomposition. The control is via the ratio of the smallest to the largest singular values, which is the reciprocal of an estimated condition number for the input matrix. If this ratio is smaller than the default threshold LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLzYkUSMxMkYnRjIvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy, the matrix is deemed to be ill-conditioned, and the least-squares calculation is based on the singular value decomposition. This default threshold is modified with the svdtolerance parameter.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0lDUmVjaXByb2NhbEVzdGltYXRlZENvbmRpdGlvbk51bWJlckc2Il0zQzY5QkYxNEE3RDM3MEUw

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEqaW5mb2xldmVsRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiUtRiw2JVErU3RhdGlzdGljc0YnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmZuLUkjbW5HRiQ2JFEiNUYnRkAtRkk2LVEiOkYnRkBGTEZORlBGUkZURlZGWEZaRmduLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjBlbUYnLyUmZGVwdGhHRmVvLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZhbzYmRmNvRmZvRmlvL0ZccFElYXV0b0YnRjotRkk2LVEjOi1GJ0ZARkxGTkZQRlJGVEZWRlgvRmVuRmhvL0ZobkZoby1GLDYlUSpMaW5lYXJGaXRGJ0YvRjItRjY2JC1GIzYsLUY2NiYtRiM2Ki1GLDYlUSJ4RidGL0YyLUZJNi1RIixGJ0ZARkwvRk9GMUZQRlJGVEZWRlhGZXAvRmhuUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGZXEtRiw2JVEiekYnRi9GMkZlcS1GLDYlUSJ3RidGL0YyRkBGQEZCRkVGZXEtRiw2JVEmZXZhbGZGJ0YvRjItRjY2JC1GIzYkLUYsNiVRI0JkRidGL0YyRkBGQEZlcUZkci1GNjYkLUYjNiQtRjY2Ji1GIzYqLUZqbjYkUSIxRidGQEZlcS1Gam42JFEiMkYnRkBGZXEtRmpuNiRRIjNGJ0ZARmVxLUZqbjYkUSI0RidGQEZARkAvRkNRJyZsYW5nO0YnL0ZGUScmcmFuZztGJ0ZARkBGZXFGXnFGQEZARj1GQA==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

In LinearFit (algebraic form)

SVD tolerance set to .10e-11 confidence level set to .95

Warning, model is not of full rank

rank = final value of residual sum of squares: 26.3923271806008

LCoqJl0zRjgyMTE3RkRBOUIwQTY4IiIiSSJ4RzYiRiVGJSomXTNGNzBGQUQ2NDhGNUU1NkNGJUkieUdGJ0YlISIiKiZdM0Y2RUZGQzU4QTYzMTIyNkYlSSJ6R0YnRiVGKyomXTNGOEVGQTM4NzkzNEYxRTRGJUkid0dGJ0YlRiU=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

In LinearFit (algebraic form)

SVD tolerance set to .1e-16 confidence level set to .95

Warning, model is not of full rank

rank = final value of residual sum of squares: 18.6547034958714

LCoqJl00MkU0MTgxODdFMzQ3QjNBIiIiSSJ4RzYiRiVGJSomXTQyQjc0RkVEMTc4MjEyRTVGJUkieUdGJ0YlISIiKiZdNDJCMzJCQ0FGMEU0NTE1QkYlSSJ6R0YnRiVGKyomXTQyREEzQ0IzNjRBQTdGNEFGJUkid0dGJ0YlRis=

JSFH

The reciprocal of the estimated condition number is slightly larger than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLzYkUSMxN0YnRjIvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy, but that is well below the default threshold of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLzYkUSMxMkYnRjIvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy, so the first least-squares calculation is based on the singular value decomposition; in the second where the reciprocal of the estimated condition number is slightly larger than the threshold, the calculation is based on the default QR decomposition, and consequently fails.

JSFH

<Text-field style="Heading 3" layout="Heading 3"><Equation executable="false" style="Heading 3" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPkYnRi8vJSdpdGFsaWNHRjEvRjNRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTS1GLDYlUSJjRidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YvRjlGOw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEickYnLyUlYm9sZEdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2L1EiPkYnRi8vJSdpdGFsaWNHRjEvRjNRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTS1GLDYlUSJjRidGL0YyLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJ0YvRjlGOw==</Equation></Text-field>

<Text-field style="Heading 4" layout="Heading 4">Full Rank</Text-field>

JSFH

Define the full-rank matrix

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

JSFH

for which

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI0NmRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIk

JSFH

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 15.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNDZkYnRi9GMkY5, and 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.

JSFH

That the system is consistent can be seen from

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRjY2Ji1GIzYnLUYsNiVRI0NmRidGL0YyLUkjbW9HRiQ2LVEifGdyRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlgtRjY2Ji1GIzYtLUkjbW5HRiQ2JFEiMUYnRkUtRkI2LVEiLEYnRkVGRy9GS0YxL0ZNRklGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWlEsMC4zMzMzMzMzZW1GJy1Gam42JFEjMzBGJ0ZFRl1vRmluRl1vLUZqbjYkUSY1MDA5MUYnRkUtRkI2LVEiL0YnRkVGR0ZKRkxGTkZQRlJGVC9GV1EsMC4xNjY2NjY3ZW1GJy9GWkZgcC1Gam42JFEmMTMzMTlGJ0ZFLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRmVwRkVGRUZncEZqcEZlcEZFRkVGZXBGRQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNTlPZTctM1duVz0=

JSFH

Consequently, this is not a least-squares problem, but a properly determined system with a unique solution, obtainable for example, by LinearSolve in LinearAlgebra.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEsTGluZWFyU29sdmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNDZkYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2Ji1GIzYtLUkjbW5HRiQ2JFEiMUYnRkFGPS1GZ242JFEjMzBGJ0ZBRj1GZm5GPS1GZ242JFEmNTAwOTFGJ0ZBLUY+Ni1RIi9GJ0ZBRkMvRkdGRS9GSUYxRkpGTEZORlAvRlNRLDAuMTY2NjY2N2VtRicvRlZGZm8tRmduNiRRJjEzMzE5RidGQS8lK2V4ZWN1dGFibGVHRkVGQUZBLyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZbcEZBRkFGW3BGQQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1dWZaOS0zV25XPQ==

JSFH

<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

JSFH

Example 16.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNDZkYnRi9GMkY5, and 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.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

That the system is inconsistent can be seen from

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNW1QZTctM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Because the matrix is full-rank, the null space is empty, and the least-squares solution is unique.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1I1F3V0AhM1duVz0=

evalf[10]( );

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1PWxaOS0zV25XPQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In general, the sum-of-squares of residuals is given by 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.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IyItNGVAIlIxIyIsXXouKDRG

evalf[10]( );

JCIrUVh1O3chIio=

JSFH

Alternatively, the solution can also be found with the LSSolve command in the Optimization package.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

The first member of the output list is half the sum-of-squares of the residuals; doubling this number gives

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrUVh1O3chIio=

JSFH

<Text-field style="Heading 4" layout="Heading 4">Rank-Deficient</Text-field>

JSFH

Define the rank-deficient matrix

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

JSFH

for which

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRI0NkRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZA

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIj

JSFHJSFH

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Example 17.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNDZEYnRi9GMkY5, and 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.

JSFH

That the system is consistent can be seen from

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRjY2Ji1GIzYnLUYsNiVRI0NkRidGL0YyLUkjbW9HRiQ2LVEifGdyRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRlgtRjY2Ji1GIzYtLUkjbW5HRiQ2JFEkMTAxRidGRS1GQjYtUSIsRidGRUZHL0ZLRjEvRk1GSUZORlBGUkZUL0ZXUSYwLjBlbUYnL0ZaUSwwLjMzMzMzMzNlbUYnRmluRl1vLUZCNi1RKiZ1bWludXMwO0YnRkVGR0ZKRmFvRk5GUEZSRlQvRldRLDAuMjIyMjIyMmVtRicvRlpGam8tRmpuNiRRIzc4RidGRUZdb0Zmby1Gam42JFEjNzdGJ0ZFLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRmJwRkVGRUZkcEZncEZicEZFRkVGYnBGRQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNT1SZTctM1duVz0=

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General solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1TXRaOS0zV25XPQ==

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Minimum-norm solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1eXlaOS0zV25XPQ==

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The general solution of an overdetermined but consistent system can also be found with the LinearSolve command from LinearAlgebra.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1OSF5V0AhM1duVz0=

JSFH

It is left to the reader to show that by appropriately redefining the free parameter in one general solution, the other will be obtained.

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<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

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Example 18.

Solve the least-squares problem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ1RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2J1EidkYnRjhGOkY8Rj5GRA==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSNDZEYnRi9GMkY5, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1JKG1mZW5jZWRHRiQ2Ji1GIzYqLUkjbW5HRiQ2JFEiMUYnRj4tRjs2LVEiLEYnRj5GQC9GQ0YxRkRGRkZIRkpGTC9GT1EmMC4wZW1GJy9GUlEsMC4zMzMzMzMzZW1GJy1GWTYkUSIyRidGPkZmbi1GWTYkUSIzRidGPkZmbi1GWTYkUSI0RidGPkY+Rj4vJSVvcGVuR1EnJmxhbmc7RicvJSZjbG9zZUdRJyZyYW5nO0YnRj4=.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

That the system is inconsistent can be seen from

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciNXFTZTctM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

General solution:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1QSV5V0AhM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Minimum-norm solution:

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1JXB5V0AhM1duVz0=

evalf[10]( );

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1WSl5V0AhM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Numeric linear algebra:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1YSN6V0AhM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

This calculation fails because the default QR-based method does not recognize that the rank-deficiency of the matrix. The more robust SVD-based method must be invoked.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1RSZ6V0AhM1duVz0=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The LSSolve command from the Optimization package can find only a local solution, that is, one member of the general solution family.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

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

Project this solution onto the row space of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQ2RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn

The columns of N are a basis for the row space; P projects onto the row space.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

The projection is the minimum-norm solution.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1MS1bOS0zV25XPQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

In this example, the LinearFit command from the Statistics package finds the minimum-norm solution, but this outcome is dependent on the relative values of the default setting of the svdtolerance parameter, and the reciprocal of the approximate condition number computed for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQ2RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVErU3RhdGlzdGljc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzotRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRiw2JVEqTGluZWFyRml0RidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNi0tRlM2Ji1GIzYpLUYsNiVRInhGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZKL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGaG4tRiw2JVEiekYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmhuLUYsNiVRJmV2YWxmRidGL0YyLUZTNiQtRiM2JS1GLDYlUSNDZEYnRi9GMkZkb0Y5RjlGaG5GXHAtRlM2JC1GIzYlLUZTNiYtRiM2Ky1JI21uR0YkNiRRIjFGJ0Y5RmhuLUZfcTYkUSIyRidGOUZobi1GX3E2JFEiM0YnRjlGaG4tRl9xNiRRIjRGJ0Y5RmRvRjlGOS9GZ29RJyZsYW5nO0YnL0Zqb1EnJnJhbmc7RidGZG9GOUY5RmhuRldGZG9GOUY5RmRvRjk=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Warning, model is not of full rank

LCgqJl0zRjg3MDk5NkE4QjNGNkRDIiIiSSJ4RzYiRiVGJSomXTNGQTNFMUY4QTYyRUNEMTNGJUkieUdGJ0YlRiUqJl0zRjdCMjVFN0NGQ0Y4NkVFRiVJInpHRidGJSEiIg==

The rank-deficiency of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQ2RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn had been detected, and the SVD-based method invoked. The minimum-norm solution is returned.

The reciprocal of the approximate condition number of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjQ2RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1VS5bOS0zV25XPQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

L0k6UmVjaXByb2NhbENvbmRpdGlvbk51bWJlckc2Il0wMDAwMDAwMDAwMDAwMDAw

This value is smaller than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21vR0YkNi1RKiZ1bWludXMwO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLzYkUSMxMkYnRjIvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy, the default svdtolerance parameter, so the more robust SVD-based method is invoked.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Nonlinear Multivariate Fit</Text-field>

<Text-field style="Heading 3" layout="Heading 3">Overdetermined Case</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The first two columns of the matrix

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

are the abscissas and ordinates, respectively, of five data points 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. The numbers in the third column are five corresponding observations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRi82JVEiZkYnRjJGNS1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiUtRi82JVEieEYnRjJGNUY4Rj0tRkE2LVEiLEYnRkRGRi9GSkY0RktGTUZPRlFGUy9GVlEmMC4wZW1GJy9GWVEsMC4zMzMzMzMzZW1GJy1GLDYlLUYvNiVRInlGJ0YyRjVGOEY9RkRGREZE.

Example 19.

Fit the function 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 to the data in M.

Since

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIk1GJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIk

, these data points generate a set of overdetermined nonlinear equations that are necessarily inconsistent. In contrast to the linear case, there is no functionality for obtaining a least-squares fit for nonlinear equations. The tools of Statistics and Optimization are the only ones that apply.

Solution

Specify the nonlinear model: define the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

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

JSFH

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

Apply the NonlinearFit command from Statistics

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQsJCooXTNGRjQ5RjBFNUFBQzU2OEYiIiIpSSJ4RzYiXTNGRjNDQzk5QzMwMUQ5MTRGJilJInlHRildM0ZEODhEN0E2RjM4MkFCOUYmRiYkIitnJWVUbiohIzY=

Apply the LSSolve command from Optimization

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjA7MSpII3pxJFshIzs3JS9JImFHNiJdM0ZGNDlGMEU1QUFDNTY4Ri9JImJHRildM0ZGM0NDOTlDMzAxRDkxNC9JImNHRildM0ZEODhEN0E2RjM4MkFCOQ==

Half the sum of squares is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+. Double it to get the minimized LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

JCIrZyVlVG4qISM2

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Apply the Minimize command from Optimization

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyQkIjMjKUhbKCpmJWVUbiohIz43JS9JImFHNiJdM0ZGNDlGMEUxRjhCODJFNi9JImJHRildM0ZGM0NDOTlERUIxQTAyOC9JImNHRildM0ZEODhEN0E1RjU5QzFFRQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The results from all three approaches are fairly consistent.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 3" layout="Heading 3">Underdetermined Case</Text-field>

<Text-field style="Heading 4" layout="Heading 4">Consistent</Text-field>

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The first two columns of the matrix

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Ki1GLDYlUSJNRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEjOj1GJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JKG1hY3Rpb25HRiQ2JC1GIzYoLUYjNilGKy1GVTYkLUkobWZlbmNlZEdGJDYoLUYjNigtSSdtdGFibGVHRiQ2Ni1JJG10ckdGJDYoLUkkbXRkR0YkNigtSSNtbkdGJDYkUScyLjI0NjdGJ0Y+LyUpcm93YWxpZ25HRi4vJSxjb2x1bW5hbGlnbkdGLi8lK2dyb3VwYWxpZ25HRi4vJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGYXAtRmNvNigtRmZvNiRRJzUuMjIxOUYnRj5GaW9GW3BGXXBGX3BGYnAtRmNvNigtRmZvNiRRJzYuNTYyMkYnRj5GaW9GW3BGXXBGX3BGYnBGaW9GW3BGXXAtRmBvNigtRmNvNigtRmZvNiRRJzIuMDA4M0YnRj5GaW9GW3BGXXBGX3BGYnAtRmNvNigtRmZvNiRRJzYuMDY1NkYnRj5GaW9GW3BGXXBGX3BGYnAtRmNvNigtRmZvNiRRJzYuMzI2MUYnRj5GaW9GW3BGXXBGX3BGYnBGaW9GW3BGXXAvJSZhbGlnbkdRJWF4aXNGJy9Gam9RKWJhc2VsaW5lRicvRlxwUSdjZW50ZXJGJy9GXnBRJ3xmcmxlZnR8aHJGJy8lL2FsaWdubWVudHNjb3BlR0Y2LyUsY29sdW1ud2lkdGhHUSVhdXRvRicvJSZ3aWR0aEdGXHMvJStyb3dzcGFjaW5nR1EmMS4wZXhGJy8lLmNvbHVtbnNwYWNpbmdHUSYwLjhlbUYnLyUpcm93bGluZXNHUSVub25lRicvJSxjb2x1bW5saW5lc0dGZ3MvJSZmcmFtZUdGZ3MvJS1mcmFtZXNwYWNpbmdHUSwwLjRlbX4wLjVleEYnLyUqZXF1YWxyb3dzR0ZCLyUtZXF1YWxjb2x1bW5zR0ZCLyUtZGlzcGxheXN0eWxlR0ZCLyUlc2lkZUdRJnJpZ2h0RicvJTBtaW5sYWJlbHNwYWNpbmdHRmRzLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGQi8lKXJlYWRvbmx5R0Y2LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGPkY+L0krbXNlbWFudGljc0dGJFEnTWF0cml4RicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGZHUvJSthY3Rpb250eXBlR1EucnRhYmxlYWRkcmVzc0YnRmp0Rl11Rl91RmF1Rj4tRiM2Jy1GOzYtUSI6RidGPkZARkNGRUZHRklGS0ZNRk9GUkZddS9GYnVRKDJEfk1hdGhGJ0Y+RmR1Rl11RmV2Rj5GZHVGXXZGanRGXXVGX3VGYXVGPkYrRl11Rj4=

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are the abscissas and ordinates, respectively, of two data points 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. The numbers in the third column are two corresponding observations 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.

JSFH

Example 20.

Since the data generate a set of two equations in three unknown parameters, this is an interpolation problem in which

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVElUmFua0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIk1GJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQEY9RkA=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

IiIj

suggests there will be a general solution with one free parameter. In the nonlinear case, there is no theory by which a (unique) minimum-norm solution is extracted.

Solution

Specify the nonlinear model by defining LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

From the two given data points, form two equations in the three unknown parameters.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYyLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RidGPi8lJmZlbmNlR0ZCLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmduLUYvNiVRImZGJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlLUYvNiVRIk1GJ0YyRjUtRiM2KUY6LUZKNi1RIixGJ0Y+Rk0vRlBGNEZRRlNGVUZXRlkvRmZuUSYwLjBlbUYnL0ZpblEsMC4zMzMzMzMzZW1GJ0Y6RjJGQEZDRjVGRkZpby1GLDYlRmRvLUYjNilGOkZpby1GOzYkUSIyRidGPkYyRkBGQ0Y1RkZGQEY+Rj4tRko2LVEiPUYnRj5GTUZPRlFGU0ZVRldGWUZlbkZobi1GLDYlRmRvLUYjNilGOkZpby1GOzYkUSIzRidGPkYyRkBGQ0Y1RkYtRko2LVEiO0YnRj5GTUZccEZRRlNGVUZXRllGXXBGaG4tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGXnAvJSZkZXB0aEdGanEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRiw2JUYuLUYjNidGZXBGMkZARkNGNUZGRklGam4tRl5vNiQtRiM2KC1GLDYlRmRvLUYjNihGZXBGaW9GOkZARkNGPkZGRmlvLUYsNiVGZG8tRiM2KEZlcEZpb0ZlcEZARkNGPkZGRkBGQ0Y+Rj5GaHAtRiw2JUZkby1GIzYoRmVwRmlvRl9xRkBGQ0Y+RkZGQEY+

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LyooSSJhRzYiIiIiKSQiJm5DIyEiJUkiYkdGJUYmKSQiJj5BJkYqSSJjR0YlRiYkIiZBYydGKg==

LyooSSJhRzYiIiIiKSQiJiQzPyEiJUkiYkdGJUYmKSQiJmMxJ0YqSSJjR0YlRiYkIiZoSydGKg==

Solve two equations for any two parameters in terms of the third. Here, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the free parameter.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCQvSSJiRzYiLCYqJiQiK1gsaHllISIqIiIiLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiYkIisoPXpRXyIhIzVGK0kiYUdGJUYrRitGKyEiIiomJCIrK1o9IVImRipGKy1GLTYjLCQqJiQiK2FGdiFlIkY2RitGN0YrRitGK0YrL0kiY0dGJSwmKiYkIispKnB2UkVGKkYrRjxGK0Y4KiYkIitOVCVSRiNGKkYrRixGK0Yr

The general solution is a fitting function dependent on one free parameter:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkZHNiI2JEkieEdGJUkieUdGJSoqSSJhR0YlIiIiKUYnIyIqUnM+ZSYiKisrKysmRispRiosJiomJCIrXVdEJSlbISM1RistSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjRidGKyEiIiomJCIrSSdHImVPRjVGKy1GNzYjRihGK0Y8RispRigjIiooSChbImYiKysrKys1Ris=

Numeric solutions that seek to minimize a sum-of-squares of residuals return, at best, individual members of this family of solutions.

JSFH

<Text-field style="Heading 4" layout="Heading 4">Inconsistent</Text-field>

JSFH

Example 21.

Fit 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 to the two points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtbkdGJDYkUScyLjI0NjdGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYkUSc1LjIyMTlGJ0Y0RjctRjE2JFEnNi41NjIyRidGNEY0RjRGNA== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtbkdGJDYkUScyLjI0NjdGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIsRidGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdRJXRydWVGJy8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYkUSc1LjIyMTlGJ0Y0RjctRjE2JFEnNi4zMjYxRidGNEY0RjQtSSNtaUdGJDYjUSFGJ0Y0.

The data determines two inconsistent equations in the three unknown parameters 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. This is no longer an interpolation; it is a least-squares problem.

Solution

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

Define P and Q, the two data points.

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

Form SS, the sum of squares of residuals.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYoLUYsNiVRImZGJ0YvRjItRlM2JC1GIzYnLUklbXN1YkdGJDYlLUYsNiVRIlBGJ0YvRjItRiM2Ji1JI21uR0YkNiRRIjFGJ0Y5Ri8vJStleGVjdXRhYmxlR0Y9RjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUZpbjYlRltvLUYjNiYtRmFvNiRRIjJGJ0Y5Ri9GZG9GMkZmb0Zkb0Y5RjktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZccS1GaW42JUZbby1GIzYmLUZhbzYkUSIzRidGOUYvRmRvRjJGZm9GZG9GOUY5RmNwLyUxc3VwZXJzY3JpcHRzaGlmdEdGaG8tRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSEZbcUZdcS1GUDYlLUZTNiQtRiM2KUZXLUZTNiQtRiM2KC1GaW42JS1GLDYlUSJRRidGL0YyLUYjNiZGYG9GZG8vJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRidGOUZmb0Zpby1GaW42JUZmci1GIzYmRmVwRmRvRltzRjlGZm9GZG9GW3NGOUY5RmhwLUZpbjYlRmZyLUYjNiZGYnFGZG9GW3NGOUZmb0Zkb0Zbc0Y5RjlGYHNGZXEtRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GZG9GOQ==

Form and solve the three normal equations.

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

PCUvSSJhRzYiRiQvSSJiR0YlRicvSSJjR0YlLCYqJiQiK2FWNl1nISM1IiIiLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiYkIisrKzpXayEiKkYvRiQhIiJGL0YvRi8qJiQiK0YwTSgqW0YuRi9GJ0YvRjs=

The general solution is a fitting function dependent on two free parameters:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

Ly1JIkZHNiI2JEkieEdGJUkieUdGJSooSSJhR0YlIiIiKUYnSSJiR0YlRispRigsJiomJCIrYVY2XWchIzVGKy1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComJCIrKys6V2shIipGK0YqISIiRitGK0YrKiYkIitGME0oKltGM0YrRi1GK0Y/Ris=

Numeric solutions that seek to minimize LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjU1NGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn return, at best, individual members of this family of solutions.

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